apr7 iir fir filters

8/6/2019 Apr7 Iir Fir Filters

1/147

M o t o r o l a s H i g h - P e r f o r m a n c e D S P T e c h n o l o g y

APR7/DRev. 2

Motorola's

DSP56000/SPS/DSP56001

Implementing

IIR/FIR Filterswith

Digital SignalProcessors


2/147

MOTOROLA APR 7

MotorolaDigital SignalProcessors

Implementing IIR/FIRFilters with MotorolasDSP56000/DSP56001

byJohn Lane and Garth HillmanDigital Signal Processing Division


3/147

Motorola reserves the right to make changes without further notice to any products here-in. Motorola makes no warranty, representation or guarantee regarding the suitability ofits products for any particular purpose, nor does Motorola assume any liability arising outof the application or use of any product or circuit, and specifically disclaims any and allliability, including without limitation consequential or incidental damages. Typical pa-rameters can and do vary in different applications. All operating parameters, includingTypicals must be validated for each customer application by customers technical ex-perts. Motorola does not convey any license under its patent rights nor the rights of oth-ers. Motorola products are not designed, intended, or authorized for use as components

in systems intended for surgical implant into the body, or other applications intended tosupport or sustain life, or for any other application in which the failure of the Motorolaproduct could create a situation where personal injury or death may occur. Should Buyerpurchase or use Motorola products for any such unintended or unauthorized application,Buyer shall indemnify and hold Motorola and its officers, employees, subsidiaries, affili-ates, and distributors harmless against all claims, costs, damages, and expenses, andreasonable attorney fees arising out of, directly or indirectly, any claim of personal injuryor death associated with such unintended or unauthorized use, even if such claim alleg-es that Motorola was negligent regarding the design or manufacture of the part. Motorolaand B are registered trademarks of Motorola, Inc. Motorola, Inc. is an Equal Opportunity/Affirmative Action Employer.

Motorola Inc. 1993


4/147

Tableof Contents

MOTOROLA iii

SECTION 1

Introduction

1.1 Analog RCL Filter Types 1-4

1.2 Analog Lowpass Filter 1-5

1.3 Analog Highpass Filter 1-91.4 Analog Bandstop Filter 1-9

1.5 Analog Bandpass Filter 1-12

SECTION 2

Second-OrderDirect-Form IIR

Digital FilterSections

2.1 Digital Lowpass Filter 2-9

2.2 Digital Highpass Filter 2-11

2.3 Digital Bandstop Filter 2-18

2.4 Digital Bandpass Filter 2-18

2.5 Summary of Digital Coefficients 2-22

SECTION 3

Single-SectionCanonic Form

(Direct Form II)

3.1 The Canonic-Form DifferenceEquation 3-1

3.2 Analysis of Internal Node Gain 3-5

3.3 Implementation on the DSP56001 3-10

SECTION 4

Single-SectionTranspose Form

4.1 Gain Evaluation of Internal Nodes 4-1

4.2 Implementation on the DSP56001 4-2

SECTION 5

Cascaded DirectForm

5.1 Butterworth Lowpass Filter 5-1

5.2 Cascaded Direct-Form Network 5-9


5/147

iv MOTOROLA

Tableof Contents

SECTION 6

Filter Design

and AnalysisSystem

6.1 Canonic Implementation 6-2

6.2 Transpose Implementation(Direct Form I) 6-14

SECTION 7

FIR FILTERS

7.1 Linear-Phase FIR Filter Structure 7-2

7.2 Linear-Phase FIR Filter DesignUsing the Frequency SamplingMethod 7-8

7.3 FIR Filter Design Using FDAS 7-227.4 FIR Implementation on the

DSP56001 7-29

REFERENCES Reference-1


6/147

Illustrations

MOTOROLA v

Figure 1-1

Figure 1-2

Figure 1-3

Figure 1-4

Figure 1-5

Figure 1-6

Figure 1-7

Figure 1-8

Figure 2-1

Figure 2-2

s-Domain Analysis of Second-Order LowpassAnalog Filter 1-6

Gain and Phase Response of Second-OrderLowpass Analog Filter at Various Valuesof Damping Factor, d 1-7

s-Domain Analysis of Second-Order HighpassAnalog Filter 1-10

Gain and Phase Response of Second-OrderHighpass Analog Filter at Various Valuesof Damping Factor, d 1-11

s-Domain Analysis of Second-Order BandstopAnalog Filter 1-13

Gain and Phase Response of Second-OrderBandstop Analog Filter at Various Values ofDamping Factor, d 1-14

s-Domain Analysis of Second-Order BandpassAnalog Filter 1-15

Gain and Phase Response of Second-OrderBandpass Analog Filter at Various Values of

Damping Factor, d 1-16

Spectrum of Bandlimited Signal Repeated atMultiples of the Sampling Frequency, fs 2-3

Direct-Form Implementation of Second-OrderLowpass IIR Filter and Analytical FormulasRelating Desired Response to Filter Coefficients 2-12


7/147

vi MOTOROLA

IllustrationsFigure 2-3

Figure 2-4

Figure 2-5

Figure 2-6

Figure 2-7

Figure 2-8

Figure 2-9

Figure 2-10

Figure 2-11

Figure 2-12

Gain and Phase Response of Second-OrderLowpass IIR Filter 2-13

DSP56001 Code and Data Structures forSecond-Order Direct-Form Implementationof a Lowpass IIR Filter 2-14

Direct-Form Implementation of Second OrderHighpass IIR Filter and Analytical FormulasRelating Desired Response to Filter Coefficients 2-15

Gain and Phase Response of Second-OrderHighpass IIR Filter 2-16

DSP56001 Code and Data Structures forSecond-Order Direct-Form Implementation

of a Highpass IIR Filter 2-17

Direct-Form Implementation of Second-OrderBandstop IIR Filter and Analytical FormulasRelating Desired Response to Filter Coefficients 2-19

DSP56001 Code and Data Structures forSecond-Order Direct-Form Implementationof a Bandstop IIR Filter 2-20

Gain and Phase Response of Second-OrderBandstop IIR Filter 2-21

Direct-Form Implementation of Second-OrderBandpass IIR Filter and Analytical FormulasRelating Desired Response to Filter Coefficients 2-23

DSP56001 Code and Data Structures forSecond-Order Direct-Form Implementationof a Bandpass IIR Filter 2-24


8/147


9/147


10/147


11/147

x MOTOROLA

IllustrationsFigure 7-9

Figure 7-10

Figure 7-11

Figure 7-12

Window Function Effects on Filter Example 7- 21

FDAS Output for FIR Bandpass Filter Examplewith a Kaiser Window 7-23

FDAS Output for FIR Bandpass Filter Example

with Equiripple Design 7-28

FIR Filter Example 7-31


12/147

MOTOROLA 1-1

SECTION 1

SECTION 1This application note considers the design of frequen-cy-selective filters, which modify the frequency content

and phase of input signals according to some specifi-

cation. Two classes of frequency-selective digital filtersare considered: infinite impulse response (IIR) and fi-

nite impulse response (FIR) filters. The design process

consists of determining the coefficients of the IIR or FIR

filters, which results in the desired magnitude and

phase response being closely approximated.

Therefore, this application note has a two-fold purpose:

1. to provide some intuitive insight into digital filters,

particularly how the coefficients are calculated in

the digital domain so that a desired frequency

response is obtained, and

2. to show how to implement both classes of digital

filters (IIR and FIR) on the DSP56001.

It is assumed that most readers are analog designers

learning digital signal processing (DSP). The ap-

proach used reflects this assumption in that digital

filters are initially presented from an analog point of

view. Hopefully, this approach will simplify the transi-

tion from the analog s-domain transfer functions to the

Introduction

Two classes of

frequency-

selective digital

filters are

considered:infinite impulse

response (IIR)

and finite

impulse

response (FIR)

filters.


13/147

1-2 MOTOROLA

equivalent functions in the digital z-domain. In

keeping with this analog perspective, IIR filters will

be discussed first since the equivalent of FIR filtersare infrequently encountered in the analog world.

SECTION 1 is a brief review of lowpass, highpass,

bandpass, and bandstop analog filters. The s-domain

formulas governing the key characteristics, magni-

tude-frequency response, G(), and phase-frequency response,

(

), are derived from first

principals. Damping factor, d, cutoff frequency, c,for lowpass and highpass filters, center frequency,

0, for bandpass and bandstop filters, and qualityfactor, Q, are defined for the various filter types.

SECTION 2 introduces the bilinear transformation so

that analog s-domain designs can be transformed

into the digital z-domain and the correct coefficients

thereby determined. The form of the formulas for the

z-domain filter coefficients thus determined are gen-

eralized in terms of the key filter characteristics in the

z-domain so that the engineer can design digital fil-

ters directly without the necessity of designing the

analog equivalent and transforming the design back

into the digital domain.

In the analog domain, the performance of the filter

depends on the tolerance of the components. Simi-

larly, in the digital domain, the filter performance is

limited by the precision of the arithmetic used to im-

plement the filters. In particular, the performance of

digital filters is extremely sensitive to overflow, which


14/147

MOTOROLA 1-3

occurs when the accumulator width is insufficient to

represent all the bits resulting from many consecu-

tive additions. This condition is similar to thecondition in the analog world in which the signal out-

put is larger than the amplifier power supply so that

saturation occurs. A short analysis of the gain at

critical nodes in the filters is given in SECTION 3

and SECTION 4 to provide some insight into the

scaling requirements for different forms of IIR filters.

For this reason, the signal flow graphs developed

are centralized about the accumulator nodes.

The analysis of IIR filters focuses on second-order

sections. Clearly, higher order filters are often re-

quired. Therefore, a brief discussion of how second

order sections can be cascaded to yield higher order

filters is given in SECTION 5. Because the analysis

becomes complex quickly, the discussion naturallyleads to using commercially available filter design

software such as Filter Design and Analysis System

(FDAS) from Momentum Data Systems, Inc. SEC-

TION 6 concludes by showing how the filter

coefficients just discussed can be used in DSP56000

code to implement practical digital filters. Examples

of complete filter designs are given, including the

code, coefficients, frequency response, and maxi-

mum sample frequency.

FIR filters are discussed in SECTION 7. Initially, FIR

filters are contrasted with IIR filters to show that in

many ways they are complementary, each satisfy-

ing weaknesses of the other. An intuitive approach


15/147

1-4 MOTOROLA

is taken to calculating the filter coefficients by start-

ing from a desired arbitrary frequency response.

The importance of and constraint imposed by linearphase is emphasized. Having developed an intuitive

appreciation of what FIR filter coefficients are, the use

of FDAS to accelerate the design process is de-

scribed. SECTION 7 concludes by showing how the

filter coefficients just determined can be used in

DSP56000 code to implement practical digital filters.

An example of a passband digital filter using a Kaiser-

window design approach is presented.

1.1 Analog RCL Filter TypesIn the following paragraphs, the analog RCL filter

network will be analyzed for the four basic filter

types: lowpass, highpass, bandpass, and band-

stop. Analyzing analog filter types shows that

designing digital IIR filters is, in many cases, much

simpler than designing analog filters.

In this analysis, as in all of the following cases, the

input is assumed to be a steady-state signal con-

taining a linear combination of sinusoidal

components whose rms (or peak) amplitudes are

constant in time. This assumption allows simple an-

alytic techniques to be used in determining the

network response. Even though these results will

then be applied to real-world signals that may not

satisfy the original steady-state assumption, the de-


16/147

MOTOROLA 1-5

viation of the actual response from the predicted

response is small enough to neglect in most cases.

General analysis techniques consist of a linearcombination of steady-state and transient re-

sponse solutions to the differential equations

describing the network.

1.2 Analog Lowpass FilterThe passive RCL circuit forming a lowpass filter

network is shown in Figure 1-1 where the transfer

function, H(s), is derived from a voltage divider

analysis of the RCL network. This approach is valid

since the effect of C and L can be described as a

complex impedance (or reactance, Xc and XL) un-

der steady state conditions; s is a complex variableof the complex transfer function, H(s). The filter fre-

quency response is found by evaluating H(s) with

s = j, where = 2f and f is the frequency of a si-nusoidal component of the input signal. The output

signal is calculated from the product of the input sig-

nal and H(j). To facilitate analysis, the input andoutput signal components are described by the

complex value, ejt = cos t + j sin t. The actualphysical input and output signal components are

found by taking the real part of this value. The input

is R{ejt} = cos t; the output is R{H(j)ejt} = G()cos [t + ()]. The previous technique is basedupon the solution of the differential equations de-

scribing the network when the input is steady state.

Describing the circuit response by H(s) instead of


17/147

1-6 MOTOROLA

solving the differential equation is a common simpli-

fication used in this type of analysis.


18/147

MOTOROLA 1-7

V0

Vi

-----X

C||R

XL

||R-------------=

=R/jC( )/ R 1+ /jC( )

jL R/jC( )/ R 1/jC+( )+--------------------------------------------------------------

=1/LC

2

j+ /RC 1/LC+-------------------------------------------------

LC R

V0Vi

=c2

2

jdc

c2+ +

-------------------------------------------

Let s = j and define H(s) = Vo/Vi; then,

Xc = 1/jC

XL = jL

c =

d =

1

LC-----------

L

R2

C

------------

Figure 1-1 s-Domain Analysis of Second-Order Lowpass Analog Filter

H s( ) 1

s/c

( )2 d s/c

( )+ 1+---------------------------------------------------=

which is the s-domain transfer function. The gain, G(), of the filter is:

G ( ) H s( )H s( )s j=

1

1 2/

c2

( )2

d/c

( )2+----------------------------------------------------------------=

where denotes complex conjugate.The phase angle, , is the angle between the imaginary and

( ) tan 1 l H s( ){ }/R H s( ){ }[ ]

tan=1 d /c( )

1 /c( )2

-------------------------------

tan= 1d /c( )

1 /c( )2

-------------------------------

for c

for > c

real components of H(s).


19/147

1-8 MOTOROLA

The magnitude of H(s) is defined as the gain, G(),

Figure 1-2 Gain and Phase Response of Second-Order Lowpass Analog

Filter at Various Values of Damping Factor, d

Gain

0.1 1.0 10Normalized Frequency (/c)

1.5

1.0

0.5

0

0.1 1.0 10

0

-/2

-

Phase

d=1.41

d=2.0

d=0.707

d=1.0

Normalized Frequency (/c)

d=1.41d=2.0

d=0.707d=1.0


20/147

MOTOROLA 1-9

of the system; whereas, the ratio of the imaginary

part to real part of H(s), I{H(j)}/R{H(j)}, is the tan-

gent of the phase, (), introduced by the filter. Ifthe input signal is Ak sin (kt + K), then the outputsignal is AKG(K) sin [Kt + K+ (K)] Figure 1-2shows the gain, G(), and phase, (), plots for thesecond-order lowpass network of Figure 1-1 for var-

ious values of damping factor, d; d also controls the

amplitude and position of the peak of the normal-

ized response curve.

The frequency corresponding to the peak amplitude

can be easily found by taking the derivative of G()(from the equation for G() in Figure 1-1) with re-spect to and setting it equal to zero. Solving theresultant equation for then defines M as the fre-quency where the peak amplitude occurs. The peak

amplitude is then GM = G(M):

Eqn. 1-1

Eqn. 1-2

for d < . For d > , M = 0 is the position of thepeak amplitude where GM = 1. When d = , GM = 1,

which gives the maximally flat response curve used in

the Butterworth filter design (usually applies only to a

set of cascaded sections). Note that C for a lowpassfilter is that frequency where the gain is G(c) = 1/dand the phase is

(

c) = -/2.

M

c

1 d2/2( )=

GM1

d 1 d2/4( )

----------------------------------=

2 2

2


21/147

1-10 MOTOROLA

1.3 Analog Highpass Filter

The passive RCL circuit forming a highpass filternetwork is shown in Figure 1-3 where the transfer

function, H(s), is again derived from a voltage divid-

er analysis of the RCL network. The gain and phase

response are plotted in Figure 1-4 for different val-

ues of damping coefficient. As evidenced, the

highpass filter response is the mirror image of the

lowpass filter response.

1.4 Analog Bandstop Filter

The analog RCL network for a bandstop filter net-

work is simply the sum of the lowpass and highpass

transfer functions shown in Figure 1-5 where the

transfer function, H(s), is again derived from a volt-age divider analysis of the RCL network. The gain

and phase response are plotted in Figure 1-6 for dif-

ferent values of quality factor, Q, (where Q = 1/d).

Neglecting the departure of real RCL components'

values from the ideal case, the attenuation at the

center frequency, f0, is infinite. Also, note that the

phase undergoes a 1 80-degree shift when passing

through the center frequency (zero in the s-plane).

Q for bandpass and bandstop filters is a measure

of the width, , of the stopband with respect tothe center frequency, O, i.e., = Q

-1o. ismeasured at the points where G() = 1/ .2


22/147

MOTOROLA 1-11

V

0Vi

-----

X

L

||R

XC

XL

+ ||R-------------------------=

=jLR/ jL R+( )

1/jC jLR+ /jL R+--------------------------------------------------------

=

2

2

j+ /RC 1/LC+-------------------------------------------------

C

L R

V0

Vi

= 2

2jd

c

c2+ +

-------------------------------------------

Let s = j and define H(s) = Vo/Vi; then,

Xc

= 1/jC

XL = jL

c =

d =

1

LC-----------

L

R2

C

------------

H s( )s/

c( )2

s/c

( )2 d s/c

( )+ 1+---------------------------------------------------=

which is the s-domain transfer function. The gain, G(), of the filter is:

G ( ) H s( )H s( ) s j=/

c( )2

1 2/

c2

( )2

d/c

( )2+----------------------------------------------------------------=

where denotes complex conjugate.The phase angle, , is the angle between the imaginary and

( ) tan 1 l H s( ){ }/R H s( ){ }[ ]

tan= 1 d /c( )1 /c( )

2

-------------------------------

tan=1 d /c( )

1 /c( )2

-------------------------------

for c

for > c

Figure 1-3 s-Domain Analysis of Second-Order Highpass Analog Filter

real components of H(s).


23/147

1-12 MOTOROLA

Figure 1-4 Gain and Phase Response of Second-Order Highpass Analog Filter

at Various Values of Damping Factor, d

Gain

0.1 1.0 10Normalized Frequency (/c)

1.5

1.0

0.5

0

0.1 1.0 10

/2

0

Phase

d=1.414d=2.0

d=0.707d=1.0

d=1.414d=2.0

d=0.707d=1.0

Normalized Frequency (/c)


24/147


25/147


26/147

MOTOROLA 1-15

Figure 1-6 Gain and Phase Response of Second-Order Bandstop Analog Filter

at Various Values of Damping Factor, d

Gain

0.1 1.0 10Normalized Frequency (/0)

1.0

0.5

0

0.1 1.0 10

/2

0

-/2

Phase

d=1.414

d=2.0

d=0.707d=1.0

d=1.414d=2.0

d=0.707d=1.0

Normalized Frequency (/0)


27/147


28/147


29/147


30/147

MOTOROLA 2-1

SECTION 2

The traditional approach to deriving the digital filtercoefficients has been to start with the digital z-domaindescription, transform to the analog s-domain to un-

derstand how to design filters, then transform back to

the digital domain to implement the filter. This ap-

proach is not used in this report. Instead, formulas are

developed relating the s-domain filter to the z-domain

filter so transformations to and from one domain to the

other are no longer necessary.

The Laplace or s-transform in the analog domain was

developed to facilitate the analysis of continuous time

signals and systems. For example, using Laplace

transforms the concepts of poles and zeros, making

system analysis much faster and more systematic.

The Laplace transform of a continuous time signal is:

Eqn. 2-3

where: L = the Laplace transform operator and

implies the operation described inEqn. 2-3

X s( ) L x t( ){ }=

x t( )e st td0

Second-OrderDirect-Form IIRDigital Filter Sections

In the direct-

formimplementation,

the aiand biare

used directly in

the difference

equation, which

can be easily

programmed on

a high-speed

DSP such as the

DSP56001.


31/147


32/147


33/147


34/147


35/147


36/147


37/147


38/147

MOTOROLA 2-9

Eqn. 2-18 can be directly implemented in software,

where x(n) is the sample input and y(n) is the corre-

sponding filtered digital output. When the filteroutput is calculated using Eqn. 2-18, y(n) is calcu-

lated using the direct-form implementation of the

digital filter.

There are other implementations which can be

used for the same system (filter) transfer function,

H(z). The canonic-form implementation and thetranspose-form implementation are discussed in

subsequent sections. First, the directform imple-

mentation will be applied to the transfer function,

H(s), developed in SECTION 1Introduction.

2.6 Digital Lowpass FilterUsing the analog transfer function, H(s), from Fig-

ure 1-1, Eqn. 2-13 and Eqn. 2-14, the digital transfer

function, H(z), becomes that shown in Figure 2-10,

where the coefficients , , and are expressed interms of the digital cutoff frequency, c, and thedamping factor, d. The value of the transfer function

at = c in the digital domain is identical to the valueof the s-domain transfer function at = c:

Eqn. 2-19

As shown in Figure 2-11, the digital gain and

phase response calculated from the equations of

Hz

ejc

( ) Hs

jc

( )=


39/147


40/147

MOTOROLA 2-11

The DSP56001 code to implement the second-or-

der lowpass filter section is shown in Figure 2-12.

The address register modifiers are initially set toM0 = 4, M4 = 1, and M5 = 1 to allow use of the cir-

cular buffer or modulo addressing in this particular

implementation (see Reference 8). Typically, this

code would be an interrupt routine driven by the input

data (A/D converter, for example) sample rate clock.

The basic filter code and the interrupt overhead and

data l/O moves for this second-order filter could be

executed at a sample rate of nearly 1 MHz on theDSP56001.

2.7 Digital Highpass FilterThe highpass filter is nearly identical to the lowpass

filter as shown by the formulas in Figure 2-13. As

with the analog case, the digital highpass filter is

just the mirror image of the lowpass filter (see Fig-

ure 2-14). The frequency transformation from high

to low in the analog case is l/; whereas, in thedigital case, it is /.

The DSP56001 code is shown in Figure 2-15; as

seen by comparison to the code shown in Figure 2-

12, the same instruction sequence is used. The only

difference is the coefficient data, which is calculated

by the formulas given in Figure 2-13. The scaling

mode is turned on so that a move from the A or B

accumulator to the X or Y register or to memory re-

sults in an automatic multiply by two. The scaling

mode is used not only in the code for the lowpass


41/147


42/147


43/147


44/147


45/147


46/147


47/147


48/147

MOTOROLA 2-19

2.8 Digital Bandstop Filter

The formulas and network diagram for the digitalbandstop filter are presented in Figure 2-16. The

DSP56001 code from Figure 2-17 is identical to that

for the lowpass and highpass cases except for the

coefficient data calculated from the equations of

Figure 2-16. Scaling of this filter is not a problem for

the singlesection case since the gain from the

equation in Figure 2-16 never exceeds unity (as is

true in the analog case as seen by the gain equationfrom Figure 1-5). Figure 2-18 is the calculated gain

and phase of the digital filter, which should compare

to the response curves of the equivalent analog fil-

ter plotted in Figure 1-6.

2.9 Digital Bandpass Filter

Because there is one less coefficient in the bandpass

network (see Figure 2-19), one instruction can be

saved in the DSP56001 code implementation shown

in Figure 2-20. Otherwise, the instructions are identi-

cal to those in the other three filter routines. Like the

second-order bandstop network, the maximum re-

sponse at the center frequency, 0, is unity for anyvalue of Q so that scaling need not be considered in

the implementation of a single-section bandpass fil-

ter. This is true when the formulas for , , and (from Figure 2-19) are used in the direct-form imple-

mentation in Figure 2-20. Figure 2-21 is the

calculated gain and phase of the digital filter, which

should compare to the response curves of the equiv-

alent analog filter plotted in Figure 1-8.


49/147

2-20 MOTOROLA

Transfer Function

Network Diagram

H z( ) 1 2 cos0z

1z

2+( )

1/2 z 1 z 2+---------------------------------------------------------=

GainG ( )

cos ccos

d 0sinsin( )2

4 0coscos( )2

+[ ]1/2

--------------------------------------------------------------------------------------------------=

Phase

( ) tan 12 0coscos( )

d 0sinsin-----------------------------------------=

Coefficients

12--- 1 tan

0/2Q( )

1 tan 0/2Q( )+--------------------------------------=

1/2 +( ) 0cos=

1/2 +( )/2=

2

z-1

z-1

y(n)

z-1

z-1

x(n)

2cos0

d2tan 0/2Q( )

sin0-------------------------------=

Figure 2-16 Direct-Form Implementation of Second-Order Bandstop IIR Filter and

Analytical Formulas Relating Desired Response to Filter Coefficients


50/147


51/147


52/147


53/147


54/147


55/147


56/147

MOTOROLA 2-27

Figure 2-22 Summary of Digital Coefficients for the Four Basic Filter Types

Difference Equation (Direct Form)

Z-Domain Transfer Function

H z( ) 1 z1 z 2+ +( )

1/2 z 1 z 2+-------------------------------------------------=

Coefficients

12---

1 1/2dsin01 1/2dsin0+-----------------------------------= 1/2 +( ) 0cos=d

2tan 0/2Q( )0sin

--------------------------------=

y n( ) 2 x[ n( ) x n 1( ) x n 2( ) ] y n 1( ) y n 2( )+ + +{ }=

where 0 < < 1/2 andQ

00------

2f0/fs2 f2 f1( )/fs---------------------------------

f0f2 f1---------------= = =

where f0 is the center frequency of the bandpass or bandstop filter,f1 and f2 are the half-power points (where gain is equal to 1/ ), andfs is the sample frequency. Note the f0 is replaced with fc in the lowpassand highpass cases.

2

Lowpass

HighpassBandpassBandstop

(1/2 + )/4(1/2 + +)/4(1/2 - )/2(1/2 + )/2

2

-20

-2cos0

1

1-11

Type Unity Gain at

f = 0

f = fs/2f = f0f = 0 and f = fs/2

NOTE: 0 = 2f0/fs

Numerator Coefficients


57/147


58/147


59/147


60/147

MOTOROLA 3-3

The diagram of Figure 3-24(b) represents the same

transfer function implemented by Eqn. 3-20, but

now the delay variable is w(n). Comparison of thisnetwork with that of the direct-form network of Fig-

ure 3-24 (a) shows that interchanging the order of

the left and right halves does not change the overall

system response (see Reference 11).

The delay elements can then be collapsed to pro-

duce the final canonic-form network shown in Figure

3-24 (c). As a result, the memory requirements for

the system are reduced to the minimum (two loca-

tions); therefore, this realization of the IIR filter is

often referred to as the canonic form. The system

difference equations for the canonic realization are:

Eqn. 3-21a

where:

Eqn. 3-2b

To prove that Eqn. 3-21a and Eqn. 3-2b are equiv-

alent to Eqn. 3-20, the following procedure can be

used. First, combine Eqn. 3-21a and Eqn. 3-2b:

Eqn. 3-3

y n( ) bi

w n i( )( )i 0=

2

=

w n( ) x n( ) ajj 1=

2

w n j( )( )=

y n( ) bii 0=

2

x n i( ) ajw n j i( )j 1=

2

=

= bix n i( )

i 0=

2

aj biw n i j( )i 0=

2

j 1=

2

= bix n i( )

i 0=

2

ajy n j( )j 1=

2


61/147


62/147

MOTOROLA 3-5

cular nature of twos-complement arithmetic).

To insure that overflow does not occur, it is necessary

to calculate the gain at the internal nodes (accumula-tor output) of a filter network. Although canonic

realization is susceptible to overflow, it is advanta-

geous because of the minimum storage requirements

and implementation in fewer instruction cycles.

3.12 Analysis of InternalNode Gain

Calculating the gain at any internal node is no differ-

ent than calculating the total network gain. In Figure

3-25, the transfer function of node w(n) is Hw(z).

Hw(z) represents the transfer function from the input

node, x(n), to the internal node, w(n). The gain,

Gw(), at w(n) is found in the standard manner byevaluating the magnitude of Hw(z) as shown in Fig-

ure 3-26. Note that the flow diagram of Figure 3-25

uses the automatic scaling mode (multiply by a fac-

tor of two when transferring data from the

accumulator to memory) so that the coefficients are

by definition less than one. The peak gain, go, of

Gw() is found by taking the derivative of Gw() withrespect to and setting the result equal to zero. Asimple expression for go is derived as follows:

Eqn. 3-5go

1

2---

psin------------------------------= for

1

2--- +

2

2---------------------

ocos 1


63/147


64/147

MOTOROLA 3-7

y(n)x(n) +

+

+

z-1

z-1

+

w(n-2)

2

2

W(z)w(n)

w(n-1)

w(n-2)

Figure 3-25 Internal Node Transfer Function, Hw(z), of Canonic

(Direct Form II) Network

X(z) Y(z)

(Left Shift)(Left Shift)

W(z)

2---

2---

H z( ) Y z( )X x( )------------=

Hw

z( ) Y z( )X z( )------------=

H z( ) 1 z1 z 2++( )

1

2--- z1

z2

+

-------------------------------------------------= Hw z( )

1

2--- z 1 z 2+--------------------------------------=

w z )( )z

--------------

w z )( )

z

2--------------


65/147


66/147


67/147


68/147


69/147


70/147


71/147


72/147


73/147


74/147


75/147


76/147


77/147


78/147

MOTOROLA 5-1

. . . the

cascaded direct-

form networkbecomes

canonic as the

filter order N

increases.

SECTION 5

By placing any of the direct-form second-order filternetworks from Figure 2-10, Figure 2-13, Figure 2-16,

and Figure 2-19 in series (i.e., connect the y(n) of oneto the x(n) of the next), a cascaded filter is created.

The resulting order N of the network is two times the

number of second-order sections. An odd-order net-

work can be made simply by adding one first-order

section in the chain. In general, to achieve a particular

response, the filter parameters associated with each

second-order section are different, since generating a

predefined total response requires that each sectionhas a different response. This fact becomes more ob-

vious when discussing the special case of Butterworth

lowpass filters.

5.16 Butterworth Lowpass FilterThe Butterworth filter response is maximally flat in the

passband at the expense of phase linearity and steep-

ness of attenuation slope in the transition band. For

lowpass or highpass cascaded second-order sections,

all sections have the same center frequency (not the

case for bandpass filters). For this reason, it is easy to

design since all that remains to be determined are the

Cascaded Direct Form


79/147


80/147


81/147

5-4 MOTOROLA

Using ,Eqn. 5-7 becomes:

Eqn. 5-10

where: pk= pk1 and

pk* = pk2(complex conjugate of pk)

Eqn. 5-10 is useful in that the response of the sys-

tem can be analyzed entirely by studying the poles

of the polynomial. However, for purposes of trans-

forming to the z-domain, Eqn. 5-7 can be used as

previously shown in Figure 2-10.

To examine the gain and phase response (physically

measurable quantities) of the lowpass Butterworth

filter, the transfer function, H(s), of Eqn. 5-7 will be

converted into a polar representation. The magni-

tude of H(s) is the gain, G(); the angle between thereal and imaginary components of H(s), (), is thearctangent of the phase shift introduced by the filter:

Eqn. 5-11

H s( ) 1s/c( ) pk[ ] s/c( ) pk[ ]

---------------------------------------------------------------------

k 1=

N/2

=

G ( ) H s( )H s( )s j==

1

/c

( )2

12

dk

/c

( )2

+

---------------------------------------------------------------------------------

k 1=

N/2

=


82/147


83/147


84/147


85/147


86/147

MOTOROLA 5-9

set of the center of each section reduces the final

center response, which must be compensated at

some point in the filter network. For cases such ashigher order Butterworth bandpass filter designs,

commercially available filter design packages such

as FDAS are useful. The use of FDAS is discussed

in SECTION 6Filter Design And Analysis Sys-

tem (FDAS) and SECTION 7Fir Filters.

5.17 Cascaded Direct-

Form NetworkFigure 5-29 shows the cascaded direct-form net-

work and data structures for the DSP56001 code

implementation of Figure 5-30. By cascading net-

work diagrams presented in SECTION 2 Second-

Order Direct-Form IIR Digital Filter Sections, theset of delays at the output of one section can be

combined with the set of delays at the input of the

next section, thus reducing the total number of de-

lays by almost a factor of two. For this reason, the

cascaded direct-form network becomes canonic as

the filter order N increases. The DSP56001 code

(see Figure 5-30) shows an example of reading

data from a user-supplied memory-mapped analog-

to-digital converter (ADC) and writing it to a memo-

ry-mapped digital-to-analog converter (DAC). The

number of sections (nsec) in these examples is

three; thus, the filter order is six. The total instruction

time for this filter structure is 600 nsec + 800 ns, in-

cluding the data l/O moves (but excluding the

interrupt overhead).


87/147


88/147


89/147


90/147


91/147


92/147


93/147


94/147


95/147


96/147

MOTOROLA 6-7

Filter Type Low Pass

Analog Filter Type ButterworthPassband Ripple In -dB -.5000Stopband Ripple In -dB -20.0000

Passband Cutoff Frequency 200.000 HertzStopband Cutoff Frequency 300.000 HertzSampling Frequency 1000.00 HertzFilter Order: 6

Filter Design Method: Bilinear Transformation

Coefficients of Hd(z) are Quantized to 24 bits

Quantization Type: Fixed Point FractionalCoefficients Scaled For Cascade Form II

NORMALIZED ANALOG TRANSFER FUNCTION T(s)

Numerator Coefficients Denominator Coefficients***************************************** **********************************S**2 TERM S TERM CONST TERM S**2 TERM S TERM CONST TERM

.000000E+00 .000000E+00 .100000E+01 .100000E+01.193185E+01 .100000E+01

.000000E+00 .000000E+00 .100000E+01 .100000E+01.141421E+01 .100000E+01

.000000E+00 .000000E+00 .100000E+01 .100000E+01.517638E+01 .100000E+01INITIAL GAIN 1.00000000

UNNORMALIZED ANALOG TRANSFER FUNCTION T(s)

Numerator Coefficients Denominator Coefficients

***************************************** **********************************

S**2 TERM S TERM CONST TERM S**2 TERM S TERM CONST TERM.000000E+00 .000000E+00 .299809E+07 .100000E+01.334500E+04 .299809E+07.000000E+00 .000000E+00 .299809E+07 .100000E+01.244871E+04 .299809E+07

.000000E+00 .000000E+00 .299809E+07 .100000E+01.896290E+03 .299809E+07INITIAL GAIN 1.00000000

DIGITAL TRANSFER FUNCTION Hd(z)

Numerator Coefficients Denominator Coefficients

***************************************** **********************************Z**2 Z TERM CONST TERM Z**2 Z TERM CONST TERM

.2520353794 .5040707588 .252035379 1.00000 -.1463916302 .0225079060

.2364231348 .4728463888 .236423134 1.00000 -.1684520245 .1765935421

.3606387377 .7212774754 .360638737 1.00000 -.2279487848 .5921629667INITIAL GAIN .876116210

Figure 6-10 FDAS.OUT File of Example Filter for Cascaded

Canonic Implementation (sheet 1 of 2)


97/147


98/147


99/147


100/147

MOTOROLA 6-11

394041

;

42

;multip

le

shift

left

macro

4344

mshl

macro

scount

45

m

if

scount

46

m

rep

#scount

47

m

asl

a

48

m

endif

49

m

endm

50

X:0000

org

x:0

51

X:0000

states

ds

2*nsec

52

Y:0000

org

y:0

53

coef

54

d

Y:0000

FE8F3

8

dc

$FE8F3B

;a(*,2)/2

=-.0112

5395

section

number

1

55

d

Y:0001

095E7

B

dc

$095E7B

;a(*,1)/2

=

.0731

9582

section

nunber

1

56

d

Y:0002

10215

9

dc

$102159

;b(*,2)/2

=

.1260

1769

section

number

1

57

d

Y:0003

2042B

2

dc

$2042B2

;b(*,1)/2

=

.2520

3538

section

number

1

58

d

Y:0004

D0215

9

dc

$D02159

;b(*,0)/2-0.5

=-.3739

8231

section

number

1

59

d

Y:0005

F4B2B

1

dc

$F4B2Bl

;a(*,2)/2

=-.0882

9677

section

number

2

60

d

Y:0006

0AC7E

B

dc

$OAC7EB

;a(*,1)/2

=

.0842

2601

section

number

2

61

d

Y:0007

0F218

E

dc

$OF218E

;b(*,2)/2

=

.1182

1157

section

number

2

62

d

Y:0008

1E431

3

dc

$1E431D

;b(*,1)/2

=

.2364

2319

section

number

2

63

d

Y:0009

CF218

E

dc

$CF218E

;b(*,0)/2-0.5

=-.3817

8843

section

number

2

64

d

Y:000A

DAlA0

1

dc

$DA1A01

;a(*,2)/2

=-.2960

8148

section

number

3

65

d

Y:000B

0E96B

6

dc

$OE96B6

;a(*,1)/2

=

.1139

7439

section

number

3

66

d

Y:000c

1714B

4

dc

$1714B4

;b(*,2)/2

=

.1803

1937

section

number

3

67

d

Y:000D

2E296

9

dc

$2EZ969

;b(*,1)/2

=

.3606

3874

section

number

3

68

d

Y:000E

D714B

4

dc

$0714B4

;b(*,0)/2-0.5

=-.3196

8063

section

number

3

69

Y:00c8

org

y:200

70

d

Y:00C8

38124

9

igain

dc

3674697

71

000000

scount

equ

0

;final

shift

count

72

include

'bodyZ.asm'

7374

000040

start

equ

$40

;origin

for

user

prog

ram

7576

P:0000

org

p:$0

;origin

for

reset

vec

tor

Figure6-7

COEFF.A

SMFileGeneratedbyMGENforExampleDesignand

CascadedCanonic

Implemen

tation

(Sheet2of4)


101/147


102/147


103/147

6-14 MOTOROLA

6.2 Transpose

Implementation(Direct Form I)Figure 6-8 is the output file from FDAS for a trans-

pose-form implementation. As before, it contains

information on the analog s-domain equivalent filter

as well as the final digital coefficients (listed again

in Figure 6-9), which have been properly scaled to

prevent overflow at the internal nodes and outputsof each cascaded section. Because of the stability

of the internal node gain of the transpose form and

because of the response of each second-order But-

terworth section, scaling was not done by the

program because it was not needed. Executable

code shown in Figure 6-10 is again generated by

MGEN. The code internal to each cascaded section

is seven instructions long; thus, 700 ns (assuming aDSP56001 clock of 20 MHz) is added to the execu-

tion time for each additional second-order section.


104/147


105/147


106/147


107/147

6-18 MOTOROLA

Figure6-10

COEFF.A

SMFileGeneratedbyMG

ENforExampleDesignan

dCascadedTransposeFo

rm

Implementation

(sheet1of4)

1

C0EFF

ident

1,0

2

include

'head1.asm'

3 4

page132,66,0,10

5

optcex,

mex

6

;

7

;

This

program

implements

an

IIR

filter

in

cascaded

transpose

sections

8

;

The

coefficients

of

each

section

are

scaled

for

cascaded

transpose

sections

9 1011

00FFFF

datin

equ

$ffff

;loca

tion

in

Y

memory

of

input

fi

le

12

00FFFF

datout

equ

$ffff

;loca

tion

in

Y

memory

of

output

file

13

00FFF0

m_scr

equ

$fff0

;sci

control

register

14

00FFF1

m_ssr

equ

$fff1

;sci

status

register

15

00FFF2

m_sccr

equ

$fff2

;sci

clock

control

register

16

00FFE1

m_pcc

equ

$ffe1

;port

c

control

register

17

00FFFF

m_

ipr

equ

$ffff

;inte

rrupt

priority

register

18

000140

xx

equ

@cvi(20480/(64*1.00

0))

;time

r

interrupt

value

19

FFD8F1

m_tim

equ

-9999

;boar

d

timer

interrupt

value

20

000003

nsec

equ

3

;numb

er

of

second

order

sections

21

include

'cascade1.asm'

22

;

23

;This

code

segment

implements

cascaded

biquad

sections

in

transpose

form

24

;

2526

cascade1

macro

nsec

27

m

28

m

;assumes

each

section's

coefficients

are

divided

by

2

29

m

30

m

do

#nsec,_ends

;doe

ach

section

31

m

macr

y0,yl,a

x:(r1),b

y:(r4)+,y0

;a=x(

n)*bi0/2+wi1/2,b=wi2,y0=bi1/2

32

m

asr

b

a,x0

;b=wi

2/2,x0=y(n)

33

m

mac

y0,yl,b

y:(r4)+,y0

;b=x(

n)*bi1/2+wi2/2,y0=ai1/2

34

m

macr

x0,y0,b

y:(r4)+,y0

;b=b+

y(n)*ai1/2,y0=bi2/2

35

m

mpy

y0,y1,b

b,x:(r0)+

y:(r4)+,y0

;b=x(

n)*bi2/2,save

wi1,y0=ai2

36

m

macr

x0,y0,b

x:(r0),a

a,y1

;b=b+

y(n)*ai2/2,a=next

iter

wi1,

37

m

;yl=o

utput

of

section

i


108/147


109/147

6-20 MOTOROLA

Figure6-10

COEFF.A

SMFileGeneratedbyMGE

NforExampleDesignand

CascadedTransposeForm

Implemen

tation

(sheet3of4

)

7576

P:0040

0003F8

ori

#3,mr

;disable

all

interrupts

77

P:0041

08F4B2

movep

#(xx-1

),x:m_sccr

;cd=xx

1

for

divide

by

xx

00013F

78

P:0043

08F4A1

movep

#$7,x:

m_pcc

;set

cc(2;0)

to

turn

on

timer

000007

79

P:0045

08F4B0

movep

#$2000

,x:m_scr

;enable

timer

interrupts

002000

80

P:0047

08F4BF

movep

#$c000

,x:m_

ipr

;set

interrupt

priority

for

sci

00C000

818283

P:0049

300000

move

#state

1,r0

;point

to

filter

state1

84

P:004A

310400

move

#state

2,rl

;point

to

filter

state2

85

P:004B

340000

move

#coef,

r4

;point

to

filter

coefficients

86

P:004C

0502A0

move

#nsec

l,m0

;addressing

modulo

nsec

87

P:004D

050EA4

move

#5*nse

c-l,m4

;addressing

modulo

5*nsec

88

P:004E

0502Al

move

#nsec-

1,m1

;addressing

modulo

nsec

89

P:004F

200013

clr

a

;initialize

internal

state

storage

90

P:0050

0603A0

rep

#nsec

;*

zero

statel

91

P:0051

565800

move

a,x:(r

0)+

;*

92

P:0052

0603A0

rep

#nsec

;*

zero

state2

93

P:0053

565900

move

a,x:(r

1)+

;*

9495

P:0054

F88000

move

x:(r0)

,a

y:(r4)+,y0

;a=wl

(initially

zero)

,y0=bl0/2

9697

P:0055

00FCB8

andi

#$fc,mr

;allow

interrupts

98

P:0056

0C0056

jmp

*

;wait

for

interrupt

99100

fil

ter

101

P:0057

0008F8

ori

#$08,mr

;set

scaling

mode

102

P:0058

09473F

movep

y:dati

n,yl

;get

sample

103

104

cascade1

nsec

;do

cascaded

biquads

105

+

;

106

+

;

assumes

each

section's

coe

fficients

are

divided

by

2

107

+

;


110/147

MOTOROLA 6-21

108

+

P:0059

060380

do

#nsec,_ends

;doe

ach

section

000061

109

+

P:00SB

FC81B3

macr

y0,y1,a

x:(r1),b

y:(r4)+,y0

;a=x(

n)*bi0/2+wi1/2,b=wi2,y0=bi1/

2

110

+

P:OOSC

21C42A

asr

b

a,x0

;b=wi

2/2,x0=y(n)

111

+

P:005D

4EDCBA

mac

y0,yl,b

y:(r4)+,y0

;b=x(

n)*bi1/2+wi2/2,y0=ail/2

112

+

P:00SE

4EDCDB

macr

x0,y0,b

y:(r4)+,y0

;b=b+

y(n)*ai1/2,y0=bi2/2

113

+

P:00SF

FC18B8

mpy

y0,y1,b

b,x:(r0)+

y:(r4)+,y0

;b=x(

n)*bi2/2,save

wi1,y0=ai2

114

+

P:0060

19A0DB

macr

x0,y0,b

x:(r0),a

a,y1

;b=b+

y(n)*ai2/2,a=next

iter

wi1,

115

+

;y1=o

utput

of

section

i

116

+

P:0061

FC1922

asr

a

b,x:(r1)+

y:(r4)+,y0

;a=ne

xt

iter

wi1/2,save

wi2,

117

+

;y0=n

ext

iter

bi0

118

+

_ends

119

120

P:0D62

09C73F

movep

y1,y:datout

;outp

ut

sample

121

_endp

122

123

P:0063

000004

rti

124

end

0

Errors

0

Warnings


111/147


112/147


113/147


114/147


115/147

7-4 MOTOROLA

Eqn. 7-18

The phase response, (), is found by taking the in-verse tangent of the ratio of the imaginary to real

components of G() so that:

Eqn. 7-19

where: h(i) is implicitly assumed to be real

Intuitively, it can be reasoned that, for any pulse to

retain its general shape and relative timing (i.e.,

pulse width and time delay to its peak value) after

passing through a filter, the delay of each frequency

component making up the pulse must be the same

so that each component recombines in phase to re-

construct the original shape. In terms of phase, this

implies that the phase delay must be linearly relatedto frequency or:

Eqn. 7-20

This relationship is illustrated in Figure 7-2 in which

a pulse having two components, sin t and sin 2t,passes through a linear-phase filter having a delay

of two cycles per Hz and a constant magnitude re-

sponse equal to unity. That is, the fundamental is

G ( ) h i( )eji

i 0=

N 1

=

h i( ) i jh i( ) isincos[ ]i 0=

N 1

=

( )=tan1

h i( ) isini 0=

N 1

h i( ) icosi 0=

N 1

---------------------------------------

( ) =


116/147

MOTOROLA 7-5

delayed by two cycles (4) and the first harmonic isdelayed by four cycles (8). The group delay of a

system, g, is defined by taking the derivative of ():

Eqn. 7-21

Since is the normalized frequency, g in Eqn. 7-21is a dimensionless quantity and can be related to

the group delay in seconds by dividing by the sam-

ple frequency, fs. For a linear-phase system, g isindependent of and is equal to . This fact can beseen by substituting Eqn. 7-20 into Eqn. 7-21.

In this example, g is two cycles per Hz. Note thatthe pulse shape within the filter has been retained

g

dd------

Figure 7-32 Signal Data through a FIR Filter

t

t

t

Stretched pulse within filter Composite output

pulse shape

Composite input

pulse shape

Filter

x1(t) = sin t

x2(t) = sin 2t

1 = 4-8 = -4

2 = 8-16 = -8

Legend:

Represents the unfiltered wave (i.e., no delay)

Represents the filtered wave


117/147


118/147


119/147


120/147


121/147

7-10 MOTOROLA

sides of by ej2km/N and summing over kas follows:

Eqn. 7-28

where: imis the Kronecker delta, which is equal

to one when i = m but is zero otherwise

can be used to find h(i) by simply setting i = m.

Eqn. 7-29

Eqn. 7-29 is the IDFT of the filter response. In gen-

eral, the discrete Fourier coefficients are complex;therefore, H(k) can be represented as:

where:

so that Eqn. 7-29 becomes:

H k( )ej2km Nk 0=

N 1

h i( )e j2ki Ni 0=

N 1

k 0=

N 1

= ej2 km N

h i( ) e j2ki N ej2km N

k 0=

N 1

i 0=

N 1

=

h i( )Nimi 0=

N 1

=

Nh m( )=

h i( ) 1N---- H k( )e

j2ki N

k 0=

N 1

=

H k( ) A k( )ej k( )

=

A k( ) H k( )


122/147


123/147

7-12 MOTOROLA

Eqn. 7-33

Expanding Eqn. 7-33 for even N yields:

Consider the A(k) and A(N-k) terms; if r = 0 for the

A(k) term and r = 1 for the A(N-k) term, then the ar-gument for the A(N-k) term is the negative of the

argument for the A(k) term, given that N is even.

That is:

Therefore, if A(k) = A(N-k), then all imaginary com-

ponents in Eqn. 7-33 will cancel and h(i) will be

h i( ) 1N---- A k( )e

j r k N 1( ) N 2ki N ]+[

k 0=

N 1

=

i) 1N---- A{ 0( )e

jrA 1( )e

j r N 1( ) N 2i N+[ ]++=

+A N 2( )ej r N 1( ) 2 i+[ ]

+

+A N 1( )ej r N 1( ) N 1( ) N 2 N 1( )i N+[ ]

ej N k( ) N 1( ) N 2 N k( )i N+[ ]

e

j k N 1( ) N 2 ki N[ ]e

j N 1( ) 2i+[ ]=

ej k N 1( ) N 2ki N+[ ]

ej N 2( ) 2i+[ ]

=

ej k N 1( ) N 2 ki N[ ]

=


124/147


125/147


126/147


127/147


128/147

MOTOROLA 7-17

Figure 7-34 Response Transformed from Polar Coordinates

0.5

0

-0.5

0 16 31

Re[H(k)]

Im[H(k)]

0.5

0

-0.5

0 16 31

(a) Real Part

(b) Imaginary Part

k

k


129/147

7-18 MOTOROLA

Figure 7-35 FIR Coefficients fromEqn. 7-17for Filter Example

0.2

0

-0.1

0 16 31

0.1

n

h(n)

Figure 7-36 Roots (Zeros) ofEqn. 7-36for Filter Example

Real

Imaginary


130/147


131/147


132/147


133/147


134/147


135/147


136/147

MOTOROLA 7-25

Recall that the continuous frequency response of

a FIR filter can be found by setting z = ej in the

z-transform so that:

Eqn. 7-39

If the linear-phase factor is factored out, Eqn. 7-39

can be written as:

Eqn. 7-40

Using Euler's identity, Eqn. 7-40 can be grouped

into sine and cosine terms as follows:

Eqn. 7-41

H ( ) h n( )ejn

n 0=

N 1

=

H ( ) ej N 1( ) 2

= h n( )ej N 1( ) 2 n[ ]

n 0=

N 1

ej N 1( ) 2

= {h 0( )ej N 1( ) 2

h 1( )ej N 1( ) 2 ) 1[ ]

++

h N 1( ) )e j N 1( ) 2

h N 2( )e j N 1( ) 2 ) 1[ ]}+

H ( ) e j N 1( ) 2= { h[ 0( ) h N 1( ) ] N 1( )2

------------------cos+

+j h[ 1( ) h N 2( ) ] N 12

------------- 1sin

+jh0 hN 1 N 12

-------------sin h 1( ) h N 2( ) ] N 1( )2

------------------ 1cos++


137/147

7-26 MOTOROLA

If it is assumed that linear phase is achieved by

even symmetry (i.e., h(n) = h(N-1-n) and that N is

even, reduces to:

which, with a change of variable, k = N/2-n-1, can

be written as follows:

Eqn. 7-42

which is in the following form:

Eqn. 7-43

where: A() is a real value amplitude function

P() is a linear-phase function

The linear functions, A() and P(), are to be con-trasted with the inherently nonlinear absolute value

and arctangent functions in and Eqn. 7-38. For allfour types of linear-phase filters (N even or odd and

symmetry even or odd), A() can be expressed as asum of cosines (see Reference 21).

Since an ideal filter cannot be realized, an approxima-

tion must be used. If the desired frequency response,

D(), can be specified in terms of a deviation,

, from

H ( ) ej N 1( ) 2

= 2

n 0=

N 2 1

h n( ) N 22------------- n cos

H ( ) ej N 1( ) 2

2h n( )n 0=

N 2 1

N 22------------- n cos=

H ( ) A ( )ejP=


138/147


139/147


140/147


141/147


142/147


143/147


144/147


145/147


146/147


147/147

apr7 iir fir filters

Documents